Lusztig's conjecture for finite classical groups with even characteristic

TL;DR

This paper extends the verification of Lusztig's conjecture to symplectic groups over finite fields with characteristic 2, using symmetric space theory, and provides partial results for special orthogonal groups in the same setting.

Contribution

It determines the scalars in Lusztig's conjecture for symplectic groups with characteristic 2, filling a gap in previous large-characteristic results.

Findings

Scalars for symplectic groups with p=2 are explicitly determined.

Partial results obtained for special orthogonal groups with p=2.

Application of symmetric space theory over finite fields.

Abstract

The determination of scalars involved in Lusztig's conjecture for finite reductive groups $G(F_q)$ was achieved by Waldspurger in the case of symplectic groups or orthogonal groups, under the condition that $p,q$ are large enough. Here $p$ is the characteristic of the finite field $F_q$. In this paper, we determine the scalars in the case of symplectic groups with $p = 2$, by applying the theory of symmetric spaces over a finite field due to Kawanaka and Lusztig. We also obtain a partial result in the case of special orthogonal groups with $p = 2$.